Download Electromagnetically induced transparency inside the laser cavity: Switch between first-order

PHYSICAL REVIEW A 78, 013805 共2008兲
Electromagnetically induced transparency inside the laser cavity: Switch between first-order
and second-order phase transitions
Qingqing Sun,1,2 M. Selim Shahriar,3 and M. Suhail Zubairy1,2
1
Department of Physics and Institute of Quantum Studies, Texas A&M University, College Station, Texas 77843-4242, USA
2
Texas A&M University at Qatar, Education City, P.O. Box 23874, Doha, Qatar
3
Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60208-3118, USA
共Received 22 January 2008; published 2 July 2008兲
We investigate the effect of electromagnetically induced transparency inside a laser cavity. By changing the
intensity of an external drive field, we can control the absorption to the laser field. A semiclassical analysis
shows that the system undergoes a switch between first-order and second-order phase transitions. Around the
tricritical point there could be a second-order phase transition followed by a first-order phase transition.
DOI: 10.1103/PhysRevA.78.013805
PACS number共s兲: 42.50.Gy, 68.35.Rh
I. INTRODUCTION
Lasing is a cooperative phenomenon of many atoms in a
cavity. Each atom has a dipole induced by the field from all
the other atoms; in turn its dipole also contributes to the
field, which can be uniformly treated by the mean field
theory. Although laser is a nonequilibrium system, the formal
similarity enables the analogy of its near-threshold behavior
to second-order phase transitions in the equilibrium systems
关1–3兴. For example, in the analogy to the ferromagnetic
phase transition, the electric field corresponds to the magnetization M, the unsaturated population inversion corresponds
to the temperature, and an injected field corresponds to the
external magnetic field H. The measurement of the laser “coexistence curve” and “susceptibility” below threshold confirmed this analogy quantitatively 关4兴. As a completion of
this analogy, Gatti and Lugiato showed that the correlation
length of a degenerate optical parametric oscillator diverges
when it approaches the threshold 关5兴.
Once the second order phase transition analogy was established, people started looking for similar analogy in first
order. Examples include the laser with a saturable absorber
共LSA兲 关6–8兴 and the dye laser 关9兴. Here we will concentrate
on the LSA problem. The saturable absorber is inside the
laser cavity. The absorption saturation changes the nature of
the system. So the laser output can have two stable values
and leads to bistability, hysteresis, etc. Experiments have
been done in many systems, e.g., He:Ne laser 关10兴, CO2 laser
with SF6 absorber 关11兴, and N2O laser with NH3 absorber
关12兴. In theory, Mandel and co-workers treated the problem
analytically in both semiclassical and quantum theories 关13兴.
In their model both the active cell and the absorber cell contain two-level atoms, interacting with a single cavity mode.
The relative saturability and population inversion of the two
cells determine the number of roots. Hysteresis cycle is obtained when two stable roots are available. The quantum
theory based on Fokker-Planck equation gives the field fluctuation and linewidth across the threshold. It is shown that
the finite fluctuation increases drastically when approaching
the threshold, and then goes to zero above the threshold. The
linewidth also has a sharp narrowing. In a related paper 关14兴
they showed that the inclusion of higher order derivative
terms reduces the width of the transition region, making the
1050-2947/2008/78共1兲/013805共5兲
transition threshold much sharper. This model has also been
used to study the time-dependent behavior 关15兴. Small amplitude harmonically modulated intensity and pulses corresponding to the passive Q switching are found to be the
stable solutions. Roy gave the photon distribution in this system and showed the similarity to the dye laser 关16兴.
As another concept in the phase transition theory, a tricritical point is the joint of a first-order phase transition line to
a second-order line in the parameter space. Scott pointed out
the existence of a tricritical point in his LSA model 关17兴.
Mortazavi and Singh measured the tricritical behavior by
changing the discharge current of the absorber cell 关18兴.
Later they found that the intensity fluctuation undergoes a
qualitative change through the tricritical point 关25兴. For a
second-order phase transition, the fluctuation decreases from
the thermal value to zero with increasing excitation, while
for a first-order phase transition it first increases to a superthermal value and then decreases to zero.
Systems with multistability and multicritical points have
also been explored 关19–21兴. It is shown that when the potential has higher order terms or when a multimode laser introduces more order parameters, there could be multistability
and multicritical points.
Three-level systems have also been used to model the
absorber. One example is the two-photon absorber 关22兴, in
which the middle level is assumed to be far from resonance
so that the system is very similar to a two-level system.
Agrawal considered the case of a ⌳ system 关23兴. The laser
frequency is assumed to lie midway between the ground state
sublevels. The phase transition is determined by the detuning, which can be controlled by a static electric or magnetic
field. He predicted the switch between first-order and
second-order phase transitions similar to our result, although
the mechanism is different.
We propose a system based on electromagnetically induced transparency 共EIT兲 关24兴. The laser field resonant with
one transition should have been absorbed. But at the presence of another drive field resonant with another transition
with a common level, the coherence effect renders transparency for the laser field. This property intrigues our interest in
the possible application into a LSA problem. Since the drive
field can control the absorption of the probe field, it might
simulate a system of laser with or without a saturable ab-
013805-1
©2008 The American Physical Society
PHYSICAL REVIEW A 78, 013805 共2008兲
SUN, SHAHRIAR, AND ZUBAIRY
sorber. As a result, we would be able to switch between
first-order and second-order phase transitions by simply adjusting the drive field intensity.
(a)
Gain Medium
II. FREE ENERGY FOR A LASER WITH OR WITHOUT
A SATURABLE ABSORBER
From Landau’s theory the Gibbs free energy close to a
critical point can be expanded into even powers of a displacement parameter x,
G共x兲 = C2x2 + C4x4 + C6x6 + . . . ,
共1兲
where the coefficient C2 is linear to the difference between
the reservoir variable and its critical value. The phase transition is second order if C4 is positive, while first order if C4
is negative.
A potential similar to the Gibbs free energy can be defined
for a laser 关1兴. The displacement parameter is the electric
field E. One can obtain a Fokker-Planck equation from the
laser theory and then apply it to the electric field to get an
equation of motion for the expectation value. The potential
can then be obtained by integration from
Ė = −
⳵ G共E兲
⳵E
共2兲
.
For a laser without an absorber, the equation of motion is
well known as
1
Ė = 关共A − C兲E − BE3兴,
2
Absorber
Drive
共3兲
(b)
Probe
1
2
冋冉
A−C−
冊
册
S
E − BE3 ,
1 + E2/Is
冉 冊
1
S
.
B−
8
Is
ab
Drive
c
cb
FIG. 1. 共a兲 The setup of the system. The absorber cell filled with
three-level atoms is put inside the unidirectional ring laser cavity.
共b兲 The level structure for the three-level atoms.
atoms are homogenously broadened. The incoherent pumping in the gain medium provides population inversion for the
active atoms, generating a laser field as the probe field for the
⌳-type three-level atoms in the absorber cell. This probe
field with frequency ␯ interacts with the transition 兩a典 ↔ 兩b典,
while the external drive field with frequency ␯␮ interacts
with the transition 兩a典 ↔ 兩c典. To avoid drive field recycling
we assume the mirrors to be transparent to the drive field.
For a closed system, the density matrix equations are
␳˙ bb = ⌫ab␳aa + ⌫cb␳cc −
i
共㜷abEe−i␯t␳ba − c.c.兲,
2ប
i
␳˙ cc = ⌫ac␳aa − ⌫cb␳cc − 共⍀␮e−i␾␮e−i␯␮t␳ca − c.c.兲,
2
共4兲
where S is the linear absorption from the saturable absorber
and Is is the saturation intensity. After integration the potential has a logarithmic term coming from the saturable absorber. Power series expansion gives
C4 =
ac
b
where A is the unsaturated gain in the active medium, B is
the saturation parameter, and C is the cavity loss. Both A and
B are proportional to the population inversion ␴. A simple
integration gives C4 = B / 8 which is positive. So it is a
second-order phase transition. The stable value of E changes
continuously as one move across the critical point.
For a laser with a saturable absorber inside the cavity, the
equation of motion can be modified as 关8兴
Ė =
a
共5兲
So if the saturation intensity of the absorber is small, Is
ⱕ S / B, the phase transition would be first order. The field has
a discontinuous jump from zero to some finite value when
crossing the critical point.
III. DERIVATION AND DISCUSSION
␳˙ aa = − 共⌫ab + ⌫ac兲␳aa +
␳˙ ab = − 共i␻ab + ␥ab兲␳ab −
The setup for our system is shown in Fig. 1. We use a
unidirectional ring laser cavity and assume both kinds of
013805-2
i㜷abE
2ប
i
+ ⍀␮e−i␾␮e−i␯␮t␳cb ,
2
i㜷abE
2ប
共7兲
i
共㜷abEe−i␯t␳ba − c.c.兲
2ប
i
+ 共⍀␮e−i␾␮e−i␯␮t␳ca − c.c.兲,
2
␳˙ cb = − 共i␻cb + ␥cb兲␳cb −
共6兲
共8兲
e−i␯t共␳aa − ␳bb兲
共9兲
i
e−i␯t␳ca + ⍀␮ei␾␮ei␯␮t␳ab ,
2
共10兲
PHYSICAL REVIEW A 78, 013805 共2008兲
ELECTROMAGNETICALLY INDUCED TRANSPARENCY ...
i
␳˙ ca = − 共i␻ca + ␥ca兲␳ca + ⍀␮ei␾␮ei␯␮t共␳aa − ␳cc兲
2
−
i㜷baEⴱ
2ប
ei␯t␳cb ,
D=
共11兲
␳cb = ˜␳cbe−i共␻ab−␯␮兲t,
+
+ ␥cb
冋
បD
再冏 冏
㜷abE
2ប
共2⌫ac + 4⌫cb兲 +
␳ca = ˜␳caei␯␮t .
共⌫ab + ⌫ac兲⌫cb
册冎
,
2ប
2
␥ab共⌫ab + ⌫ac兲⌫cb
⍀␮2
共⌫ab + ⌫ac − 3⌫cb + 6␥cb兲 + 2␥cb␥ca共⌫ac + 2⌫cb兲
2
冉
⍀␮2
+ ␥ab␥cb
4
2
⍀␮2
共⌫ab + ⌫cb兲 + ␥ca共⌫ab + ⌫ac兲⌫cb
2
冏 冏冋
㜷abE
冊冋
Ė =
⍀␮2
共⌫ab + ⌫cb兲 + ␥ca共⌫ab + ⌫ac兲⌫cb ,
2
1
2
冋冉
A
1 + E /Is
2
冊 冉冊 册
−C E−
␯兩㜷 兩2Ne
␯
Im P ,
⑀0
ប 2⌫ 2
⍀␮2
2
2
关2␥cb共⌫ab + ⌫ac − 3⌫cb兲 − 共⌫ab + ⌫ac兲⌫cb + 12␥cb
兴 + 2␥cb
␥ca共⌫ac + 2⌫cb兲
1 A ␯ e 兩㜷ab兩 4
C4 =
− Na
.
2
8 I s 2 4 ⑀ 0ប 3
⍀␮2
⍀␮2
4
+ ␥ab␥cb
共⌫ab + ⌫cb兲 + ␥ca共⌫ab + ⌫ac兲⌫cb
4
2
冊冋
冉
If there is no drive field, i.e., ⍀␮ = 0, it is simplified to,
C4共⍀␮=0兲 =
1 A ␯ e 兩㜷ab兩4
⌫ac + 2⌫cb
− Na
. 共17兲
2
3
8 Is 2 4⑀0ប 2␥ab共⌫ab + ⌫ac兲⌫cb
For large enough Nea we could have C4共⍀␮=0兲 ⬍ 0. Then we
increase the drive field intensity so that the second term of
Eq. 共16兲 has a smaller magnitude. Finally at some point C4
= 0. According to Ref. 关8兴 this is the criterion between firstorder and second-order phase transitions.
In the limiting case of a perfect EIT system, the dephasing
rate between the two lower levels ␥cb = 0, which also requires
⌫cb = 0. Substitute into Eqs. 共16兲 and 共17兲 and we find
C4共⍀␮⫽0兲 = A / 8Is while C4共⍀␮=0兲 → −⬁. One might tend to say
that the order of the phase transition just depends on whether
we switch on the drive field or not. But the stable value of E
would also go to infinity so it is a idealized situation. One
can never achieve a perfect EIT.
For a finite ␥cb we did the numerical calculation to find
the steady state solutions for Eq. 共15兲. These solutions will
共15兲
where A = ⑀0បab␥ab g is the linear gain and Is = 2兩㜷 ab兩2 is the satuab
ration intensity for the active atoms. After integration the
coefficient of E4 term in G共E兲 is
共13兲
where the denominator
4
册
册
and Nea is the effective number density for the three-level
atoms. It can be related to the actual number density by Nea
= NaVa / Vtot. Here Va and Vtot are the absorber volume and
the total volume, respectively. Similarly we can define the
effective number density for the gain medium Neg.
We assume the active atoms to be effectively two-level,
which has the same level separation as ␻ab and the same
dipole moment. With the polarization from the EIT medium,
we can obtain the field equation of motion
Together with the relation ␳aa + ␳bb + ␳cc = 1 for the closed
system, we can obtain the steady state solution. So the effective polarization from the absorber
兩㜷ab兩2E
4
共14兲
共12兲
P = 2Nea㜷ba˜␳ab = iNea
2ប
+
where 㜷ab is the electric dipole moment, E is the probe field
produced by the active atoms, ⍀␮ is the drive Rabi frequency, ⌫ and ␥ are the population relaxation rates and dipole decay rates, respectively.
For simplicity, we assume both the drive field and the
probe field to be resonant with their corresponding transitions. After transforming into the rotating frame,
␳ab = ˜␳abe−i␻abt,
冏 冏
㜷abE
册
共16兲
follow the change of C = Q␯ where Q is the quality factor of
the cavity. As can be seen from Fig. 2, the probe intensity-Q
curve is single valued for a Rabi frequency above the threshold, while bistable for a Rabi frequency below the threshold.
Those parts with a negative slope are unstable. So the switch
between first-order and second-order phase transitions can be
achieved by simply changing the drive Rabi frequency. For a
small range near the threshold there are even three solutions.
When the Q factor increases, one can see a second-order
phase transition followed by a first-order phase transition, as
shown for ⍀␮ = 0.1096 MHz.
The parameters we used are somehow extreme. The Q
factor and drive Rabi frequency are small. But the probe
laser intensity is large, which is still acceptable since it is
inside the cavity. There is no requirement for a small probe
because we keep E to all orders in Eq. 共13兲. An interesting
thing is to compare the threshold value to the solution of
C4 = 0. Ideally they should be the same. But we found a small
difference. This can be understood since the C4 criterion is
013805-3
PHYSICAL REVIEW A 78, 013805 共2008兲
SUN, SHAHRIAR, AND ZUBAIRY
FIG. 2. 共Color online兲 Intensity-Q factor curves for different
drive Rabi frequencies. 共a兲 From right to left, ⍀␮ = 0.100 MHz,
0.104 MHz, 0.108 MHz, 0.112 MHz, 0.116 MHz, 0.120 MHz.
Clearly there is a switch between first-order and second-order phase
transitions. 共b兲 From right to left, ⍀␮ = 0.1088 MHz, 0.1092 MHz,
0.1096 MHz, 0.1100 MHz. For a small range near the tricritical
point, there could be three nonzero solutions. The other parameters
are 兩㜷ab兩 = 10−29 m C, ␭ = 1 ␮m, Nea = 4.5⫻ 1016 m−3, Neg = 1.28
⫻ 1018 m−3, and ␥cb = 1 kHz.
only good for small probe approximation, which does not
apply quite well in this case.
The switch between first-order and second-order phase
transition is a tricritical point in the phase space, as can be
seen from the phase diagram Fig. 3. Such tricritical points
have been predicted in the two-level atom saturable absorber
case 关18兴. In that system the point is the termination of both
lines, while here the first-order line goes beyond the tricritical point. So for a given drive Rabi frequency, we can ob-
FIG. 4. 共Color online兲 Intensity-Neg curves for different drive
Rabi frequencies. From right to left, ⍀␮ = 0.011 MHz, 0.11 MHz,
0.14 MHz, 1.1 MHz. Similarly there is a switch between firstorder and second-order phase transitions. In the small window enlarged around the tricritical point, ⍀␮ = 0.121 MHz and there are
three nonzero solutions. The other parameters are 兩㜷ab兩
= 10−29 m C, ␭ = 1 ␮m, Nea = 4.5⫻ 1016 m−3, Q = 27, and ␥cb
= 1 kHz.
serve a first-order transition, or a second-order transition followed by a first-order transition, or a second-order transition.
We use the Q factor as the variable to investigate the
phase transition because it is the only parameter not in the
expression of C4. So we can compare the criterion C4 = 0 to
the numerical results. Without that purpose other parameters
may also be used as the variable, for example, the effective
number density of the gain medium. It is proportional to the
discharge current if we use a gas discharge cell as the gain
cell 关18兴. The numerical calculation shown in Fig. 4 is similar to Fig. 2. For a given drive Rabi frequency there could be
one, two, or three solutions.
The latent heat in first-order phase transition is also an
interesting topic 关8兴. At the critical point, the ClausiusClapeyron equation is
L = T0共V2 − V1兲
FIG. 3. 共Color online兲 Phase diagram for the system. The firstorder phase transition line meets the second-order line at the tricritical point and goes beyond.
dP
.
dT0
共18兲
In our system, minus threshold population inversion corresponds to the critical temperature T0, the drive field corresponds to the pressure P, and the laser field corresponds to
the volume V. E1 and E2 can be solved from Eq. 共15兲.
The LSA problem has also been investigated from the
dispersive point of view. Mandel showed that if both cells
contains two-level atoms, there could be new solutions corresponding to nonzero detuning because of the anomalous
dispersion 关26兴. Lukin et al. considered the intracavity EIT
and found a pronounced frequency pulling and cavitylinewidth narrowing 关27兴. Here we are mainly interested in
the absorption property, so we take the detuning to be zero to
avoid the pulling effect.
013805-4
PHYSICAL REVIEW A 78, 013805 共2008兲
ELECTROMAGNETICALLY INDUCED TRANSPARENCY ...
IV. CONCLUSION
In this paper, we investigate the effect of including an EIT
cell as an absorber inside the laser cavity. By controlling the
drive Rabi frequency, we can simulate the cases of laser with
or without absorber and obtain phase transitions of both first
order and second order. Around the tricritical point there
could even be a second-order phase transition followed by a
first-order one. These phenomena can be seen clearly from
the phase diagram, in which the first-order phase transition
line goes beyond its joint with the second-order line. The
关1兴
关2兴
关3兴
关4兴
关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
V. DeGiorgio and M. O. Scully, Phys. Rev. A 2, 1170 共1970兲.
R. Graham and H. Haken, Z. Phys. 237, 31 共1970兲.
H. Haken, Rev. Mod. Phys. 47, 67 共1975兲.
M. Corti and V. DeGiorgio, Phys. Rev. Lett. 36, 1173 共1976兲.
A. Gatti and L. Lugiato, Phys. Rev. A 52, 1675 共1995兲.
A. P. Kazantsev, S. G. Rautian, and G. I. Surdutovich, Zh.
Eksp. Teor. Fiz. 54, 1409 共1968兲 关Sov. Phys. JETP 27, 756
共1968兲兴; A. P. Kazantsev and G. I. Surdutovich, Zh. Eksp.
Teor. Fiz. 56, 2001 共1969兲 关Sov. Phys. JETP 29, 1075 共1969兲兴;
A. P. Kazantsev and G. I. Surdutovich, Zh. Eksp. Teor. Fiz. 58,
245 共1970兲 关Sov. Phys. JETP 31, 133 共1970兲兴.
R. Salomaa and S. Stenholm, Phys. Rev. A 8, 2695 共1973兲; R.
Salomaa, J. Phys. A 7, 1094 共1974兲.
J. F. Scott, M. Sargent III, and C. D. Cantrell, Opt. Commun.
15, 13 共1975兲.
A. Baczynski, A. Kossakowski, and T. Marszalek, Z. Phys. B
23, 205 共1976兲; R. B. Schaefer and C. R. Willis, Phys. Rev. A
13, 1874 共1976兲.
P. H. Lee, P. B. Schoefer, and W. B. Barker, Appl. Phys. Lett.
13, 373 共1968兲; H. M. Gibbs, S. L. McCall, and T. N. C.
Venkatesan, Phys. Rev. Lett. 36, 1135 共1976兲.
S. Ruschin and S. H. Bauer, Chem. Phys. Lett. 66, 100 共1979兲.
tricritical value determined by the criterion C4 = 0 is close to
the numerical result but not very accurate, because in this
case the probe field is no longer small. Our calculation is
based on ⌳-type system. Other three-level systems like V
type or ⌶ type should have a similar drive intensity controlled phase transition as well.
ACKNOWLEDGMENT
This research was partially supported by Qatar National
Research Fund 共QNRF兲.
关12兴 J. W. Won, Opt. Lett. 8, 79 共1983兲.
关13兴 L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, Phys. Rev. A 18, 238 共1978兲.
关14兴 S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, Phys. Rev. A 18, 1145 共1978兲.
关15兴 P. Mandel and T. Erneux, Phys. Rev. A 30, 1893 共1984兲; T.
Erneux and P. Mandel, ibid. 30, 1902 共1984兲.
关16兴 R. Roy, Phys. Rev. A 20, 2093 共1979兲.
关17兴 J. F. Scott, Opt. Commun. 15, 343 共1975兲.
关18兴 M. Mortazavi and S. Singh, Phys. Rev. Lett. 64, 741 共1990兲.
关19兴 P. Mandel, Z. Phys. B 33, 205 共1979兲.
关20兴 G. S. Agarwal and S. Dattagupta, Phys. Rev. A 26, 880 共1982兲.
关21兴 G. S. Agarwal, Phys. Rev. A 26, 888 共1982兲.
关22兴 P. Mandel and F.-K. Fang, Phys. Lett. 83A, 59 共1981兲.
关23兴 G. P. Agrawal, Phys. Rev. A 24, 1399 共1981兲.
关24兴 S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64,
1107 共1990兲.
关25兴 M. Mortazavi, F. Rawwagah, and S. Singh, Phys. Rev. A 65,
025803 共2002兲.
关26兴 P. Mandel, Phys. Lett. 83A, 207 共1981兲.
关27兴 M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, Opt. Lett. 23, 295 共1998兲.
013805-5